On the length of the central series (derived series) of a Moufang commutative loop
type: Item
instance of: scholarly article
title: On the length of the central series (derived series) of a Moufang commutative loop
zbMATH Open document ID: 0920.20061
DOI: 10.1007/BF02360883
author: Nicolae I. Sandu
published in: Mathematical Notes
publication date: 1999-09-20T00:00:00Z
review text: A loop \((Q,\cdot)\) is a Moufang commutative loop (MCL) if \(a^2\cdot bc=ab\cdot ac\) for all \(a,b,c\in Q\). The associator \([a,b,c]\) is defined by \(ab\cdot c=a[a,b,c]\cdot bc\). The lower (transfinite) central series of an MCL \(Q\) has the form \(Q=Q_0\supseteq Q_1\supseteq\cdots\supseteq Q_i\supseteq Q_{I=1}\supseteq\cdots\), where the normal subloop \(Q_{i+1}\) is generated by all associators \([a,u,v]\) with \(a\in Q_i\) and \(u,v\in Q\). Similarly \(Q=Q^{(0)}\supseteq Q^{(1)}\supseteq\cdots\supseteq Q^{(i)}\supseteq Q^{(i+1)}\supseteq\cdots\) is called the (transfinite) derived series of MCL \(Q\) if \(Q^{(i+1)}\) is the associant (subgroup generated by all associators) of the subgroup \(Q^{(i)}\). The least transfinite \(n\) for which \(Q_n=1\) resp. \(Q^{(n)}=1\) is called the length of the respective series. In this paper, by analogy with group theory, the author proves that there exists an MCL for which the length of the lower central series resp. derived series is equal to a prescribed transfinite number.
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 1257046
zbMATH Keywords: Moufang commutative loops
zbMATH Keywords: associators
zbMATH Keywords: central series
zbMATH Keywords: normal subloops
zbMATH Keywords: derived series
reviewed by: Elena Broží ková
MaRDI profile type: Publication
cites work: Q3248988
cites work: Free commutative Moufang loops and anticommutative graded rings
full work available at URL: https://doi.org/10.1007/bf02360883
OpenAlex ID: W1997790399
description: scientific article; zbMATH DE number 1257046